Hello, in your 1997 paper (Regarding WLS with du Toit and Spisic) you discussed the likelihood function for a binary y variable (which looks like it is bernoulli in the univariate case).
a) What happens when the categorical y variable is ordered (for example, a likert type scale)? Would we have a binomial or multinomial likelihood instead? (my guess is multinomial, corresponding to proportional odds)
b) What happens to bivariate probit regression? Does it become a binomial/multinomial probit regression? Is it even still a bivariate?
Thanks again. I have a few more questions on your previous response:
a) Can you give some intuition on why we have proportional odds model instead of cumulative probit model? Aren't we assuming normally distributed errors for the latent variables?
b) When you mention bivariate binomial, are you referring to the binomial product model? Why do you use this instead of a multinomial likelihood function for contingency tables, which is used in the Olsson 1979 paper for calculating polychoric correlations?
a) I think of prop'l odds and cumulative probit as the same, just changing the link from logit to probit. But WLSMV has only probit link.
b) With 2 binary variables you have a 2 x 2 response, but I don't treat them as one response variable, but as two binary variables obtained from a bivariate normal that is discretized. When Olsson looked at polychorics he was considering ordinal variables which is not relevant here.